Game Theory and Economics of Contracts Lecture 4 Basics in Game Theory (2)
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1 Game Theory and Economics of Contracts Lecture 4 Basics in Game Theory (2) Yu (Larry) Chen School of Economics, Nanjing University Fall 2015
2 Extensive Form Game I It uses game tree to represent the games. More clear for capturing the possible orders of events and therefore sequential games.
3 Extensive Form Game I It uses game tree to represent the games. More clear for capturing the possible orders of events and therefore sequential games. I The extensive form game contains the following information:
4 Extensive Form Game I It uses game tree to represent the games. More clear for capturing the possible orders of events and therefore sequential games. I The extensive form game contains the following information: 1. the set of players
5 Extensive Form Game I It uses game tree to represent the games. More clear for capturing the possible orders of events and therefore sequential games. I The extensive form game contains the following information: 1. the set of players 2. the order of moves
6 Extensive Form Game I It uses game tree to represent the games. More clear for capturing the possible orders of events and therefore sequential games. I The extensive form game contains the following information: 1. the set of players 2. the order of moves 3. the players payo s as a function of the moves that were made
7 Extensive Form Game I It uses game tree to represent the games. More clear for capturing the possible orders of events and therefore sequential games. I The extensive form game contains the following information: 1. the set of players 2. the order of moves 3. the players payo s as a function of the moves that were made 4. what the players choices are when they move
8 Extensive Form Game I It uses game tree to represent the games. More clear for capturing the possible orders of events and therefore sequential games. I The extensive form game contains the following information: 1. the set of players 2. the order of moves 3. the players payo s as a function of the moves that were made 4. what the players choices are when they move 5. what each player knows when he makes his choices
9 Extensive Form Game I It uses game tree to represent the games. More clear for capturing the possible orders of events and therefore sequential games. I The extensive form game contains the following information: 1. the set of players 2. the order of moves 3. the players payo s as a function of the moves that were made 4. what the players choices are when they move 5. what each player knows when he makes his choices 6. the probability distributions over any exogenous events.
10 Extensive Form Game I It uses game tree to represent the games. More clear for capturing the possible orders of events and therefore sequential games. I The extensive form game contains the following information: 1. the set of players 2. the order of moves 3. the players payo s as a function of the moves that were made 4. what the players choices are when they move 5. what each player knows when he makes his choices 6. the probability distributions over any exogenous events. I There is perfect information in such a game if each player, when making any decision, is perfectly informed of all the events that have previously occurred.
11 An n-player Extensive Game with Perfect Information 4-tuple (primitives), G = (N, H, P, (u i ) n i=1 ). Again they are common knowledge. I A set N is the set of players.
12 An n-player Extensive Game with Perfect Information 4-tuple (primitives), G = (N, H, P, (u i ) n i=1 ). Again they are common knowledge. I A set N is the set of players. I A set H of sequences ( nite or in nite) is the history, and each component of a history is an action taken by a player, if H satis es the following three properties. (1) The empty sequence? is a member of H. (2) If (a k ) k=1,...,k 2 H (where K may be in nite) and L < K then (a k ) k=1,...,l 2 H. (3) If an in nite sequence (a k ) k=1 satis es (ak ) k=1,...,l 2 H for every positive integer L then (a k ) k=1 2 H. A history h = (a k ) k=1,...,k 2 H is terminal if it is in nite or if there is no a K +1 such that (a k ) k=1,...,k +1 2 H. The set of terminal histories is denoted Z.
13 An n-player Extensive Game with Perfect Information 4-tuple (primitives), G = (N, H, P, (u i ) n i=1 ). I A function P that assigns to each nonterminal history (each member of HnZ) a member of N. (P is the player function, P(h) being the player who takes an action after the history h.)
14 An n-player Extensive Game with Perfect Information 4-tuple (primitives), G = (N, H, P, (u i ) n i=1 ). I A function P that assigns to each nonterminal history (each member of HnZ) a member of N. (P is the player function, P(h) being the player who takes an action after the history h.) I For each player i 2 N, u i : Z! R is the payo function for player i.
15 Strategies I A strategy of a player in an extensive game is a plan that speci es the action chosen by the player for every history after which it is his turn to move.
16 Strategies I A strategy of a player in an extensive game is a plan that speci es the action chosen by the player for every history after which it is his turn to move. I A n player extensive form game with perfect information Γ = (N, H, P, (u i ) n i=1 ) implicitly determine the strategies of all players.
17 Strategies I A strategy of a player in an extensive game is a plan that speci es the action chosen by the player for every history after which it is his turn to move. I A n player extensive form game with perfect information Γ = (N, H, P, (u i ) n i=1 ) implicitly determine the strategies of all players. I Let h be a history of length k; we denote by (h, a) the history of length k + 1 consisting of h followed by action a.
18 Strategies I A strategy of a player in an extensive game is a plan that speci es the action chosen by the player for every history after which it is his turn to move. I A n player extensive form game with perfect information Γ = (N, H, P, (u i ) n i=1 ) implicitly determine the strategies of all players. I Let h be a history of length k; we denote by (h, a) the history of length k + 1 consisting of h followed by action a. I After any nonterminal history h player P(h) chooses an action from the set A(h) = fa : (h, a) 2 Hg.
19 Strategies I A strategy of a player in an extensive game is a plan that speci es the action chosen by the player for every history after which it is his turn to move. I A n player extensive form game with perfect information Γ = (N, H, P, (u i ) n i=1 ) implicitly determine the strategies of all players. I Let h be a history of length k; we denote by (h, a) the history of length k + 1 consisting of h followed by action a. I After any nonterminal history h player P(h) chooses an action from the set A(h) = fa : (h, a) 2 Hg. I s i is the strategy of player i that assigns an action in A(h) to each nonterminal history h 2 HnZ for which P(h) = i
20 Example pdf I Two people intend to share two desirable identical indivisible objects. One of them proposes an allocation, which the other then either accepts or rejects. In the event of rejection, neither person receives either of the objects. Each person cares only about the number of objects he obtains.
21 Example pdf I Two people intend to share two desirable identical indivisible objects. One of them proposes an allocation, which the other then either accepts or rejects. In the event of rejection, neither person receives either of the objects. Each person cares only about the number of objects he obtains. I An extensive game that models this procedure is (N, H, P, (u i ) n i=1 ) where N = fi, 2g; H consists of the ten histories?, (2, 0), (1, 1), (0, 2), ((2, 0), y), ((2, 0), n), ((1, 1), y), ((1, 1), n), ((0, 2), y), ((0, 2), n); P(0) = 1 and P(h) = 2 for every nonterminal history h 6=?.
22 Example pdf I Z = f((2, 0), y), ((2, 0), n), ((1, 1), y), ((1, 1), n), ((0, 2), y), ((0, 2), n)g. u i is de ned by the ith number at every terminal node.
23 Example pdf I Z = f((2, 0), y), ((2, 0), n), ((1, 1), y), ((1, 1), n), ((0, 2), y), ((0, 2), n)g. u i is de ned by the ith number at every terminal node. I Player 1 takes an action only after the initial history 0, so that we can identify each of her strategies with one of the three possible actions that she can take after this history: (2, 0), (1, 1), and (0, 2); Player 2 takes an action, either y or n, after each of the three histories (2, 0), (1, 1),and (0, 2). Thus we can identify each of his strategies by specifying either y or n to be the action that he chooses after the histories. His strategies represent a contingency plan (on the history he faces when he makes a choice)
24 An equivalent representation 1 3.pdf I This suggests an alternative de nition of an extensive game in which the basic component is a tree (a connected graph with no cycles).
25 An equivalent representation 1 3.pdf I This suggests an alternative de nition of an extensive game in which the basic component is a tree (a connected graph with no cycles). I In this formulation, each node corresponds to a history, and any pair of nodes that are connected corresponds to an action; the names of the actions are not part of the de nition.
26 Example 2 I A strategy speci es the action chosen by a player for every history after which it is his turn to move, even for histories that, if the strategy is followed, are never reached.
27 Example 2 I A strategy speci es the action chosen by a player for every history after which it is his turn to move, even for histories that, if the strategy is followed, are never reached. I In this game player 1 has four strategies AE, AF, BE, and BF. That is, her strategy speci es an action after the history (A, C) even if it speci es that she chooses B at the beginning of the game.
28 Nash Equilibrium I A Nash equilibrium of an extensive game with perfect information Γ = (N, H, P, (u i ) n i=1 ) is a strategy pro le s such that for every player i we have u i (si, s i ) u i (s i, s i ) for every strategy s i of player i.
29 Nash Equilibrium I A Nash equilibrium of an extensive game with perfect information Γ = (N, H, P, (u i ) n i=1 ) is a strategy pro le s such that for every player i we have u i (s i, s i ) u i (s i, s i ) for every strategy s i of player i. I (Zermelo 1913; Kuhn 1953) A nite game of perfect information has a pure-strategy NE.
30 Example 3 I 2 NE: (A, R) and (B, L), with payo pro les (2,1) and (1,2). (B, L) is a NE because given that player 2 chooses L after the history A, it is optimal for player 1 to choose B at the start of the game (if she chooses A instead, then given player 2 s choice she obtains 0 rather than 1), and given player 1 s choice of B it is optimal for player 2 to choose L.
31 Example 3 I 2 NE: (A, R) and (B, L), with payo pro les (2,1) and (1,2). (B, L) is a NE because given that player 2 chooses L after the history A, it is optimal for player 1 to choose B at the start of the game (if she chooses A instead, then given player 2 s choice she obtains 0 rather than 1), and given player 1 s choice of B it is optimal for player 2 to choose L. I But NE (B, L) in this game lacks plausibility. If the history A were to occur then player 2 would, it seems, choose R over L, since he obtains a higher payo by doing so. The equilibrium (B, L) is sustained by the "threat" of player 2 to choose L if player 1 chooses A. This threat is not credible since player 2 has no way of committing himself to this choice.
32 Sequential Rationality and Subgames I Sequential rationality. Backward induction. The notion of equilibrium more sensible in extensive form games requires that the action prescribed by each player s strategy be optimal, given the other players strategies, after every history.
33 Sequential Rationality and Subgames I Sequential rationality. Backward induction. The notion of equilibrium more sensible in extensive form games requires that the action prescribed by each player s strategy be optimal, given the other players strategies, after every history. I The subgame of the extensive game with perfect information Γ = (N, H, P, (u i ) n i=1 ) that follows the history h is the extensive game Γ(h) = (N, Hj h, Pj h, (u i j h ) n i=1 ) where Hj h is the set of sequences h 0 of actions for which (h, h 0 ) 2 H, Pj h is de ned by Pj h (h 0 ) = P(h, h 0 ) for each h 0 2 Hj h, and u i j h is de ned by u i j h (h 0 ) u i j h (h 00 ) if and only if u i (h, h 0 ) u i (h, h 00 ), for each h 0, h 00 2 Hj h.
34 Sequential Rationality and Subgames I Sequential rationality. Backward induction. The notion of equilibrium more sensible in extensive form games requires that the action prescribed by each player s strategy be optimal, given the other players strategies, after every history. I The subgame of the extensive game with perfect information Γ = (N, H, P, (u i ) n i=1 ) that follows the history h is the extensive game Γ(h) = (N, Hj h, Pj h, (u i j h ) n i=1 ) where Hj h is the set of sequences h 0 of actions for which (h, h 0 ) 2 H, Pj h is de ned by Pj h (h 0 ) = P(h, h 0 ) for each h 0 2 Hj h, and u i j h is de ned by u i j h (h 0 ) u i j h (h 00 ) if and only if u i (h, h 0 ) u i (h, h 00 ), for each h 0, h 00 2 Hj h. I Given a strategy s i of player i and a history h in the extensive game Γ, denote by s i j h the strategy that s i induces in the subgame Γ(h) (i.e., s i j h (h 0 ) = s i (h, h 0 ) for each h 0 2 Hj h ).
35 Subgame Perfect Equilibrium I Given n player extensive form game with perfect information Γ = (N, H, P, (u i ) n i=1 ), a subgame perfect (Nash) equilibrium (SPE) is a strategy pro le s such that for every player i and every nonterminal history h 2 HnZ for which P(h) = i, u i j h (s i j h, s i j h ) u i j h (s i j h, s i j h ) for every strategy s i j h of player i in the subgame Γ(h).
36 Subgame Perfect Equilibrium I Given n player extensive form game with perfect information Γ = (N, H, P, (u i ) n i=1 ), a subgame perfect (Nash) equilibrium (SPE) is a strategy pro le s such that for every player i and every nonterminal history h 2 HnZ for which P(h) = i, u i j h (s i j h, s i j h ) u i j h (s i j h, s i j h ) for every strategy s i j h of player i in the subgame Γ(h). I Equivalently, we can de ne a subgame perfect equilibrium to be a strategy pro le s in Γ for which for any history h the strategy pro le s j h is a Nash equilibrium of the subgame Γ(h).
37 Subgame Perfect Equilibrium I Given n player extensive form game with perfect information Γ = (N, H, P, (u i ) n i=1 ), a subgame perfect (Nash) equilibrium (SPE) is a strategy pro le s such that for every player i and every nonterminal history h 2 HnZ for which P(h) = i, u i j h (s i j h, s i j h ) u i j h (s i j h, s i j h ) for every strategy s i j h of player i in the subgame Γ(h). I Equivalently, we can de ne a subgame perfect equilibrium to be a strategy pro le s in Γ for which for any history h the strategy pro le s j h is a Nash equilibrium of the subgame Γ(h). I The notion of SPE eliminates NE in which the players threats are not credible. Eg., in Example 3 the only SPE is (A, R) why?
38 Existence of SPE I (Kuhn s Theorem) Every nite extensive game with perfect information has a subgame perfect equilibrium.
39 Existence of SPE I (Kuhn s Theorem) Every nite extensive game with perfect information has a subgame perfect equilibrium. I The proof (cf. Osborne and Rubinstein s book) is constructive: for each of the longest nonterminal histories, in the game we choose an optimal action for the player whose turn it is to move and replace each of these histories with a terminal history in which the payo pro le is that which results when the optimal action is chosen; then we repeat the procedure until going back to the start of the game.
40 Existence of SPE I (Kuhn s Theorem) Every nite extensive game with perfect information has a subgame perfect equilibrium. I The proof (cf. Osborne and Rubinstein s book) is constructive: for each of the longest nonterminal histories, in the game we choose an optimal action for the player whose turn it is to move and replace each of these histories with a terminal history in which the payo pro le is that which results when the optimal action is chosen; then we repeat the procedure until going back to the start of the game. I The procedure (algorithm) used in this proof is often referred to as backwards induction. In addition to being a means by which to prove the theorem, this procedure is an algorithm for calculating the set of subgame perfect equilibria of a nite game.
41 Example 4: Stackelberg games I A Stackelberg game is a two-player extensive game with perfect information in which a "leader" chooses an action from a set A 1 and a "follower," informed of the leader s choice, chooses an action from a set A 1. The solution usually applied to such games in economics is that of SPE.
42 Example 4: Stackelberg games I A Stackelberg game is a two-player extensive game with perfect information in which a "leader" chooses an action from a set A 1 and a "follower," informed of the leader s choice, chooses an action from a set A 1. The solution usually applied to such games in economics is that of SPE. I Some (but not all) subgame perfect equilibria of a Stackelberg game correspond to solutions of the maximization problem max U 1 (a 1, a 2 ) (a 1,a 2 )2A 1 A 2 s.t.a 2 2 arg max U 2 (a 1, a2) 0 a2 0 2A 2 where U i is a payo function that represents player i s preferences. If the set A i of actions of each player i is compact and the payo functions. U i are continuous then this maximization problem has a solution.
43 Example 4: Stackelberg games I A Stackelberg game is a two-player extensive game with perfect information in which a "leader" chooses an action from a set A 1 and a "follower," informed of the leader s choice, chooses an action from a set A 1. The solution usually applied to such games in economics is that of SPE. I Some (but not all) subgame perfect equilibria of a Stackelberg game correspond to solutions of the maximization problem max U 1 (a 1, a 2 ) (a 1,a 2 )2A 1 A 2 s.t.a 2 2 arg max U 2 (a 1, a2) 0 a2 0 2A 2 where U i is a payo function that represents player i s preferences. If the set A i of actions of each player i is compact and the payo functions. U i are continuous then this maximization problem has a solution. I A concrete example.
44 Two Extensions of an extensive form Game: Exogenous Uncertainty Sometimes we need to model situations in which there is some exogenous uncertainty. An extensive game with perfect information and chance moves is a tuple (N, H, P, f c, (u i ) n i=1 ) where, as before, N is a nite set of players and H is a set of histories, and I P is a function from the nonterminal histories in H to N [ fcg.(if P(h) = c then chance determines the action taken after the history h.
45 Two Extensions of an extensive form Game: Exogenous Uncertainty Sometimes we need to model situations in which there is some exogenous uncertainty. An extensive game with perfect information and chance moves is a tuple (N, H, P, f c, (u i ) n i=1 ) where, as before, N is a nite set of players and H is a set of histories, and I P is a function from the nonterminal histories in H to N [ fcg.(if P(h) = c then chance determines the action taken after the history h. I For each h 2 H with P(h) = c, f c (jh) is a probability measure on A(h); each such probability measure is assumed to be independent of every other such measure. (f c (ajh) is the probability that a occurs after the history h.)
46 Two Extensions of an extensive form Game: Exogenous Uncertainty Sometimes we need to model situations in which there is some exogenous uncertainty. An extensive game with perfect information and chance moves is a tuple (N, H, P, f c, (u i ) n i=1 ) where, as before, N is a nite set of players and H is a set of histories, and I P is a function from the nonterminal histories in H to N [ fcg.(if P(h) = c then chance determines the action taken after the history h. I For each h 2 H with P(h) = c, f c (jh) is a probability measure on A(h); each such probability measure is assumed to be independent of every other such measure. (f c (ajh) is the probability that a occurs after the history h.) I For each player i, u i is a (expected) payo function on lotteries over the set of terminal histories (with respect to f c ).
47 Two Extensions of an extensive form Game: Exogenous Uncertainty I A strategy for each player i is de ned as before.
48 Two Extensions of an extensive form Game: Exogenous Uncertainty I A strategy for each player i is de ned as before. I The outcome of a strategy pro le is a probability distribution over terminal histories.
49 Two Extensions of an extensive form Game: Exogenous Uncertainty I A strategy for each player i is de ned as before. I The outcome of a strategy pro le is a probability distribution over terminal histories. I The de nition of a subgame perfect equilibrium is the same as before.
50 Two Extensions of an extensive form Game: Simultaneous Moves Sometimes we need to model situations in which players move simultaneously after certain histories, each of them being fully informed of all past events when making his choice. (1 principal contracting with many agents) An extensive game with perfect information and simultaneous moves (also, multi-stage game) is a tuple (N, H, P, (u i ) n i=1 ) where N, H, and u i for each i are the same as before, P is a function that assigns to each nonterminal history a set of players, and H and P jointly satisfy the condition that for every nonterminal history h there is a collection fa i (h)g i2p (h) of sets for which A(h) = fa : (h, a) 2 Hg = i2p (h) A i (h).
51 Two Extensions of an extensive form Game: Simultaneous Moves Besides, I a history in such a game is a sequence of vectors; the components of each vector a k are the actions taken by the players whose turn it is to move after the history (a l ) k l=1 1. The set of actions among which each player i 2 P(h) can choose after the history h is A i (h); the interpretation is that the choices of the players in P(h) are made simultaneously;
52 Two Extensions of an extensive form Game: Simultaneous Moves Besides, I a history in such a game is a sequence of vectors; the components of each vector a k are the actions taken by the players whose turn it is to move after the history (a l ) k l=1 1. The set of actions among which each player i 2 P(h) can choose after the history h is A i (h); the interpretation is that the choices of the players in P(h) are made simultaneously; I a strategy of player i in such a game is a function that assigns an action in A i (h) to every nonterminal history h for which i 2 P(h). The de nition of a subgame perfect equilibrium is the same as before with the exception that "P(h) = i" is replaced by "i 2 P(h)".
53 Extensive game with Imperfect Information I If a player is not able to observe all her opponents previous moves when it is her turn to move, the game has imperfect information. Things will be a lot more complicated.
54 Extensive game with Imperfect Information I If a player is not able to observe all her opponents previous moves when it is her turn to move, the game has imperfect information. Things will be a lot more complicated. I Relevant concepts: information set Belief Update Perfect Bayesian Equilibrium sender-receiver games Sequential Equilibrium.
55 Cooperative Game: Bargaining I When a buyer and a seller negotiate the price of a house they are faced with a bargaining problem. Similarly, two trading countries bargaining over the terms of trade, a basketball player discussing his contract with the owners of a team, or two corporations arguing over the details of a joint venture.
56 Cooperative Game: Bargaining I When a buyer and a seller negotiate the price of a house they are faced with a bargaining problem. Similarly, two trading countries bargaining over the terms of trade, a basketball player discussing his contract with the owners of a team, or two corporations arguing over the details of a joint venture. I In all these bargaining situations, there is usually a set S of alternative outcomes and the two sides have to agree on some element of this set. Once an agreement has been reached, the bargaining is over, and the two sides then receive their respective payo s. In case they cannot agree, the result is usually the status-quo, and we say there is disagreement.
57 Cooperative Game: Bargaining I When a buyer and a seller negotiate the price of a house they are faced with a bargaining problem. Similarly, two trading countries bargaining over the terms of trade, a basketball player discussing his contract with the owners of a team, or two corporations arguing over the details of a joint venture. I In all these bargaining situations, there is usually a set S of alternative outcomes and the two sides have to agree on some element of this set. Once an agreement has been reached, the bargaining is over, and the two sides then receive their respective payo s. In case they cannot agree, the result is usually the status-quo, and we say there is disagreement. I It is quite clear that the two sides will not engage in bargaining, unless there are outcomes in S which give both sides a higher payo than the payo s they receive from the status-quo. Thus, if (d 1, d 2 ) are the payo s from the disagreement point, then the interesting part of S consists of those outcomes which give both sides higher payo s than the disagreement payo s.
58 Bargaining Problem I A n-person bargaining problem (or game) consists of n persons (or players) indexed by i 2 N, a set S of feasible alternatives (or bargaining outcomes or simply outcomes), and a utility function u i on S for each player i, such that
59 Bargaining Problem I A n-person bargaining problem (or game) consists of n persons (or players) indexed by i 2 N, a set S of feasible alternatives (or bargaining outcomes or simply outcomes), and a utility function u i on S for each player i, such that 1. u i (s) d i for every s 2 S, where d i is the payo from the disagreement point, and
60 Bargaining Problem I A n-person bargaining problem (or game) consists of n persons (or players) indexed by i 2 N, a set S of feasible alternatives (or bargaining outcomes or simply outcomes), and a utility function u i on S for each player i, such that 1. u i (s) d i for every s 2 S, where d i is the payo from the disagreement point, and 2. at least for one s 2 S for which u i (s) > d i.
61 Bargaining Problem I A n-person bargaining problem (or game) consists of n persons (or players) indexed by i 2 N, a set S of feasible alternatives (or bargaining outcomes or simply outcomes), and a utility function u i on S for each player i, such that 1. u i (s) d i for every s 2 S, where d i is the payo from the disagreement point, and 2. at least for one s 2 S for which u i (s) > d i. I Condition (2) guarantees that there is a feasible alternative which makes all players strictly better-o relative to the disagreement point. This condition makes the bargaining problem non-trivial.
62 Bargaining Problem I A n-person bargaining problem (or game) consists of n persons (or players) indexed by i 2 N, a set S of feasible alternatives (or bargaining outcomes or simply outcomes), and a utility function u i on S for each player i, such that 1. u i (s) d i for every s 2 S, where d i is the payo from the disagreement point, and 2. at least for one s 2 S for which u i (s) > d i. I Condition (2) guarantees that there is a feasible alternative which makes all players strictly better-o relative to the disagreement point. This condition makes the bargaining problem non-trivial. I Formally, we can write a bargaining problem as a triplet B = (S, (u i, d i ) n i=1 ), where S, u 1 and u 2 satisfy properties (1) and (2).
63 Bargaining Problem I Now focus on 2-person bargaining problem for simplicity. Everything can be easily generalized to n-person cases.
64 Bargaining Problem I Now focus on 2-person bargaining problem for simplicity. Everything can be easily generalized to n-person cases. I We formally de ne a solution for a bargaining problem to be a rule (i.e., a set-valued function) that assigns to each bargaining game B a subset s(b) of the set of its outcomes S.
65 Bargaining Problem I Now focus on 2-person bargaining problem for simplicity. Everything can be easily generalized to n-person cases. I We formally de ne a solution for a bargaining problem to be a rule (i.e., a set-valued function) that assigns to each bargaining game B a subset s(b) of the set of its outcomes S. I We can think of the set s(b) as the collection of all mutually satisfactory agreements of the bargaining game B or, simply, as the solutions of the bargaining game.
66 Bargaining Problem I Now focus on 2-person bargaining problem for simplicity. Everything can be easily generalized to n-person cases. I We formally de ne a solution for a bargaining problem to be a rule (i.e., a set-valued function) that assigns to each bargaining game B a subset s(b) of the set of its outcomes S. I We can think of the set s(b) as the collection of all mutually satisfactory agreements of the bargaining game B or, simply, as the solutions of the bargaining game. I Obviously, any such rule should satisfy certain reasonable conditions.
67 Bargaining Problem Obviously, any such rule should satisfy certain reasonable conditions. I Pareto Optimality or E ciency: An outcome s 2 S is said to be Pareto optimal (or Pareto e cient) if there is no other outcome s 2 S satisfying
68 Bargaining Problem Obviously, any such rule should satisfy certain reasonable conditions. I Pareto Optimality or E ciency: An outcome s 2 S is said to be Pareto optimal (or Pareto e cient) if there is no other outcome s 2 S satisfying 1. u 1 (s) u 1 (s ) and u 2 (s) u 2 (s ), and
69 Bargaining Problem Obviously, any such rule should satisfy certain reasonable conditions. I Pareto Optimality or E ciency: An outcome s 2 S is said to be Pareto optimal (or Pareto e cient) if there is no other outcome s 2 S satisfying 1. u 1 (s) u 1 (s ) and u 2 (s) u 2 (s ), and 2. u i (s) > u i (s ) for at least one player i.
70 Bargaining Problem Obviously, any such rule should satisfy certain reasonable conditions. I Pareto Optimality or E ciency: An outcome s 2 S is said to be Pareto optimal (or Pareto e cient) if there is no other outcome s 2 S satisfying 1. u 1 (s) u 1 (s ) and u 2 (s) u 2 (s ), and 2. u i (s) > u i (s ) for at least one player i. I Intuition: This guarantees that there is no further possibility of strictly improving the utility of one of the players, while leaving the other at least as well o as she was before.
71 Bargaining Problem Obviously, any such rule should satisfy certain reasonable conditions. I Pareto Optimality or E ciency: An outcome s 2 S is said to be Pareto optimal (or Pareto e cient) if there is no other outcome s 2 S satisfying 1. u 1 (s) u 1 (s ) and u 2 (s) u 2 (s ), and 2. u i (s) > u i (s ) for at least one player i. I Intuition: This guarantees that there is no further possibility of strictly improving the utility of one of the players, while leaving the other at least as well o as she was before. I A solution rule s() is said to be Pareto optimal if for every game B the set s(b) consists of Pareto optimal outcomes.
72 Bargaining Problem Another reasonable condition. I Independence of Irrelevant Alternatives: A solution rule s is independent of irrelevant alternatives if for every bargaining game B = (S, (u 1, d 1 ), (u 2, d 2 )) and for every subset T of S satisfying (d 1, d 2 ) 2 U T = f(u 1 (s), u 2 (s)) : s 2 T g and s(b) T, we have s(b T ) = s(b), where B T is the bargaining game B T = (T, (u 1, d 1 ), (u 2, d 2 )).
73 Bargaining Problem Another reasonable condition. I Independence of Irrelevant Alternatives: A solution rule s is independent of irrelevant alternatives if for every bargaining game B = (S, (u 1, d 1 ), (u 2, d 2 )) and for every subset T of S satisfying (d 1, d 2 ) 2 U T = f(u 1 (s), u 2 (s)) : s 2 T g and s(b) T, we have s(b T ) = s(b), where B T is the bargaining game B T = (T, (u 1, d 1 ), (u 2, d 2 )). I Intuition: any acceptable bargaining solution rule should remain acceptable if we throw away alternatives that have already been considered to be less desirable by both players.
74 Bargaining Problem Last reasonable condition. I Independence of Linear Transformations: A bargaining solution s() is said to be independent of linear transformations if for any bargaining game B = (S, (u 1, d 1 ), (u 2, d 2 )), and any linear utility functions of the form v i = a i + b i u i, where the a i and b i are constants with b i > 0 for each i, the bargaining game B = (S, (v 1, b 1 d 1 + a 1 ), (v 2, b 2 d 2 + a 2 )) satis es s(b ) = s(b).
75 Bargaining Problem Last reasonable condition. I Independence of Linear Transformations: A bargaining solution s() is said to be independent of linear transformations if for any bargaining game B = (S, (u 1, d 1 ), (u 2, d 2 )), and any linear utility functions of the form v i = a i + b i u i, where the a i and b i are constants with b i > 0 for each i, the bargaining game B = (S, (v 1, b 1 d 1 + a 1 ), (v 2, b 2 d 2 + a 2 )) satis es s(b ) = s(b). I Intuition: This guarantees that the bargaining solution rule will not be a ected by changing the scale or units in which we measure utility.
76 Bargaining Problem I We now proceed to describe a solution rule which satis es all these conditions.
77 Bargaining Problem I We now proceed to describe a solution rule which satis es all these conditions. I We start by associating to each bargaining game B = (S, (u 1, d 1 ), (u 2, d 2 )) the function g B : S! R de ned by g B (s) = [u 1 (s) d 1 ][u 2 (s) d 2 ], and let σ(b) be the set of all maximizers of the function g B, i.e., σ(b) = arg max s2s g B (s).
78 Bargaining Problem I We now proceed to describe a solution rule which satis es all these conditions. I We start by associating to each bargaining game B = (S, (u 1, d 1 ), (u 2, d 2 )) the function g B : S! R de ned by g B (s) = [u 1 (s) d 1 ][u 2 (s) d 2 ], and let σ(b) be the set of all maximizers of the function g B, i.e., σ(b) = arg max s2s g B (s). I We shall call σ() the Nash solution rule and for any bargaining game B we shall refer to the members (if there are any) of σ(b) as the Nash solutions of B.
79 Remarks I Such a form of g B (s) implicitly assumes there is symmetric bargaining power between players 1 and 2. To address the asymmetric bargaining power, one may simply consider g B (s) = [u 1 (s) d 1 ] α [u 2 (s) d 2 ] β, where α, β 0, α + β = 1, and α and β will denote the weights of respective bargaining power.
80 Remarks I Such a form of g B (s) implicitly assumes there is symmetric bargaining power between players 1 and 2. To address the asymmetric bargaining power, one may simply consider g B (s) = [u 1 (s) d 1 ] α [u 2 (s) d 2 ] β, where α, β 0, α + β = 1, and α and β will denote the weights of respective bargaining power. I In such a bargaining game, the objective function to be maximized is not an individual payo but a joint (coalitional) payo. So it is a cooperative (coalitional) game.
81 Bargaining Problem I Theorem (Nash 1950) On the class of bargaining games with compact sets of utility allocations U = f(u 1 (s), u 2 (s)) : s 2 Sg, the Nash rule σ() is Pareto optimal, independent of irrelevant alternatives, and independent of linear transformations.
82 Bargaining Problem I Theorem (Nash 1950) On the class of bargaining games with compact sets of utility allocations U = f(u 1 (s), u 2 (s)) : s 2 Sg, the Nash rule σ() is Pareto optimal, independent of irrelevant alternatives, and independent of linear transformations. I Example 5: Suppose two individuals are bargaining over a sum of money; say $100. If they cannot agree on how to divide the money, none of them gets any money.
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